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| Mirrors > Home > MPE Home > Th. List > nnexpcl | Structured version Visualization version GIF version | ||
| Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 16-Dec-2005.) |
| Ref | Expression |
|---|---|
| nnexpcl | ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nnsscn 12271 | . 2 ⊢ ℕ ⊆ ℂ | |
| 2 | nnmulcl 12290 | . 2 ⊢ ((𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑥 · 𝑦) ∈ ℕ) | |
| 3 | 1nn 12277 | . 2 ⊢ 1 ∈ ℕ | |
| 4 | 1, 2, 3 | expcllem 14113 | 1 ⊢ ((𝐴 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (𝐴↑𝑁) ∈ ℕ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 (class class class)co 7431 ℕcn 12266 ℕ0cn0 12526 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: digit1 14276 nnexpcld 14284 faclbnd4lem3 14334 faclbnd5 14337 climcndslem1 15885 climcndslem2 15886 climcnds 15887 harmonic 15895 geo2sum 15909 geo2lim 15911 ege2le3 16126 eftlub 16145 ef01bndlem 16220 expgcd 16600 phiprmpw 16813 pcdvdsb 16907 pcmptcl 16929 pcfac 16937 pockthi 16945 prmreclem3 16956 prmreclem5 16958 prmreclem6 16959 modxai 17106 1259lem5 17172 2503lem3 17176 4001lem4 17181 ovollb2lem 25523 ovoliunlem1 25537 ovoliunlem3 25539 dyadf 25626 dyadovol 25628 dyadss 25629 dyaddisjlem 25630 dyadmaxlem 25632 opnmbllem 25636 mbfi1fseqlem1 25750 mbfi1fseqlem3 25752 mbfi1fseqlem4 25753 mbfi1fseqlem5 25754 mbfi1fseqlem6 25755 aalioulem1 26374 aaliou2b 26383 aaliou3lem9 26392 log2cnv 26987 log2tlbnd 26988 log2ublem1 26989 log2ublem2 26990 log2ub 26992 zetacvg 27058 vmappw 27159 sgmnncl 27190 dvdsppwf1o 27229 0sgmppw 27242 1sgm2ppw 27244 vmasum 27260 mersenne 27271 perfect1 27272 perfectlem1 27273 perfectlem2 27274 perfect 27275 pcbcctr 27320 bclbnd 27324 bposlem2 27329 bposlem6 27333 bposlem8 27335 chebbnd1lem1 27513 rplogsumlem2 27529 ostth2lem3 27679 ostth3 27682 oddpwdc 34356 tgoldbachgt 34678 faclim2 35748 opnmbllem0 37663 heiborlem3 37820 heiborlem5 37822 heiborlem6 37823 heiborlem7 37824 heiborlem8 37825 heibor 37828 dvdsexpnn0 42369 hoicvrrex 46571 ovnsubaddlem2 46586 ovolval5lem1 46667 fmtnoprmfac2lem1 47553 fmtno4prm 47562 perfectALTVlem1 47708 perfectALTVlem2 47709 perfectALTV 47710 bgoldbachlt 47800 tgblthelfgott 47802 tgoldbachlt 47803 blenpw2 48499 nnpw2pb 48508 nnolog2flm1 48511 |
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